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Harnessing the Power of Vicinity-Informed Analysis for Classification under Covariate Shift
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Kazuto Fukuchi
June 10, 2024
Research
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Harnessing the Power of Vicinity-Informed Analysis for Classification under Covariate Shift
第15回ザッピングセミナーにおける発表資料です.
Kazuto Fukuchi
June 10, 2024
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Transcript
)BSOFTTJOHUIF1PXFSPG7JDJOJUZ *OGPSNFE"OBMZTJTGPS$MBTTJ fi DBUJPO VOEFS$PWBSJBUF4IJGU ୈճβοϐϯάηϛφʔ Ұే ஜେֶཧݚ"*1 IUUQTBSYJWPSHBCT +PJOUXPSLXJUI
.JUTVIJSP'VKJLBXB 5TVLVCB3*,&/"*1 :PIFJ"LJNPUP 5TVLVCB3*,&/"*1 +VO 4BLVNB 5PLZP5FDI3*,&/"*1
ࣗݾհ w ໊લҰే 'VLVDIJ ,B[VUP w ॴଐஜେֶγεςϜใܥॿڭ w ܦྺ
w ஜେֶγεςϜใֶઐ߈Պത࢜ޙظ՝ఔमྃ w ཧݚ"*1ಛผݚڀһ w ݱࡏஜେֶγεςϜใܥॿڭ w ݱࡏཧݚ"*1٬һݚڀһ w ݚڀڵຯ w ػցֶशʹ͓͚ΔόΠΞεʢެฏੑɼసҠֶशɼҼՌਪʣ w ཧ౷ܭɼಛʹɼ൚ؔਪఆ
ࠓͷసҠֶश
సҠֶशͷશ͕ͯॻ͔Εͨຊʂ ങ͍·͠ΐ͏ʂ λΠϜ
࣍ wసҠֶश wڞมྔγϑτԼʹ͓͚Δཧղੳ w݁Ռͷৄࡉ
సҠֶश
ྨ ϥϕϧ͖σʔλ ֶशΞϧΰϦζϜ ྨث h 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ
ྨ ֶशΞϧΰϦζϜ ྨث h( )=Ҝࢠ ϥϕϧ͖σʔλ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ
ྨ ֶशΞϧΰϦζϜ ྨث h( )=Ҝࢠ ͳΔͨ͘ΔΑ͏ h Λબ͍ͨ͠ ϥϕϧ͖σʔλ 0
ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ
సҠֶश ֶशΞϧΰϦζϜ ྨث h( )=Ҝࢠ ιʔεσʔλ ༧ଌ࣌ʹҟͳΔ ੑ࣭ͷσʔλ λʔήοτ 0
ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ
సҠֶश ֶशΞϧΰϦζϜ ྨث h( )=Ҝࢠ ιʔεσʔλ ༧ଌ࣌ʹҟͳΔ ੑ࣭ͷσʔλ λʔήοτσʔλ ༧ଌ࣌ͱಉ͡ੑ࣭ͷ
σʔλΛগྔ؍ଌ ιʔεσʔλ େྔʹ֬อՄೳ λʔήοτ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ
సҠֶश ֶशΞϧΰϦζϜ ྨث h( )=Ҝࢠ ιʔεσʔλ ιʔεσʔλΛ׆༻͠ ͯΑΓߴਫ਼ͷ ༧ଌΛ࣮ݱ λʔήοτσʔλ
༧ଌ࣌ͱಉ͡ੑ࣭ͷ σʔλΛগྔ؍ଌ ιʔεσʔλ େྔʹ֬อՄೳ ༗༻ͳใΛநग़ʢసҠʣ λʔήοτ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ
సҠֶशͷޭ wྫ0 ff i DF)PNFEBUBTFU wͭͷυϝΠϯ Ξʔτ ΫϦοϓΞʔτ ϓϩμΫτ ϦΞϧ
wͷΧςΰϦ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ 1BQFSTXJUI$PEFIUUQTQBQFSTXJUIDPEFDPNTPUBEPNBJOBEBQUBUJPOPOP ff i DFIPNF ྨਫ਼
సҠֶशͷఆࣜԽɾ ཧղੳͷඪ
ྨͷֶश ֶशΞϧΰϦζϜ ྨث h( )=Ҝࢠ h ʹΑΔྨޡ͕ࠩ ࠷খʹͳΔΑ͏ʹ͢Δ ϥϕϧ͖σʔλ 0
ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ
ྨͷֶश ֶशΞϧΰϦζϜ ྨث h ʹΑΔྨޡ͕ࠩ ࠷খʹͳΔΑ͏ʹ͢Δ ϥϕϧ͖σʔλ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713
QQ h(X) = ̂ Y (X, Y) ∼ P (X, Y) iid ∼ P = (X1 , Y1 ), ⋮ , (Xn , Yn ) ྨޡࠩʢظޡࠩʣ errP (h) = 𝔼 P [1{h(X) ≠ Y}]
ʢڭࢣ͋ΓʣసҠֶश ֶशΞϧΰϦζϜ ྨث h( )=Ҝࢠ ιʔεσʔλ h ʹΑΔλʔήοτͰ ͷྨޡ͕ࠩ ࠷খʹͳΔΑ͏ʹ͢Δ
λʔήοτσʔλ λʔήοτ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ
ʢڭࢣ͋ΓʣసҠֶश ֶशΞϧΰϦζϜ ྨث h(X) = ̂ Y ιʔεσʔλ P h
ʹΑΔλʔήοτͰ ͷྨޡ͕ࠩ ࠷খʹͳΔΑ͏ʹ͢Δ λʔήοτσʔλ Q λʔήοτ Q (X, Y)P iid ∼ P = (X1 , Y1 ), ⋮ , (XnP , YnP ) (X, Y)Q iid ∼ Q = (XnP +1 , YnP +1 ), ⋮ , (XnP +nQ , YnP +nQ ) nP ≫ nQ ྨޡࠩʢظޡࠩʣ errQ (h) = 𝔼 Q [1{h(X) ≠ Y}] (X, Y) ∼ Q
ֶशཧ wߏஙͨ͠ΞϧΰϦζϜʹ͍ͭͯ༨ޡࠩͱαϯϓϧαΠζ ͷؔʢαϯϓϧෳࡶʣΛ໌Β͔ʹ͍ͨ͠
ֶशཧ wߏஙͨ͠ΞϧΰϦζϜʹ͍ͭͯ༨ޡࠩͱαϯϓϧαΠζ ͷؔʢαϯϓϧෳࡶʣΛ໌Β͔ʹ͍ͨ͠ αϯϓϧαΠζେ αϯϓϧαΠζখ ΞϧΰϦζϜ͕ग़ྗͨ͠ྨثͷޡࠩ σʔλ͕ࢁ͋Δ΄Ͳখ͘͞ͳΔʢʁʣ ༨ޡࠩ Լ͛ΒΕͳ͍ ޡࠩͷݶք
ޡࠩେ ޡࠩখ
ֶशཧ wߏஙͨ͠ΞϧΰϦζϜʹ͍ͭͯ༨ޡࠩͱαϯϓϧαΠζ ͷؔʢαϯϓϧෳࡶʣΛ໌Β͔ʹ͍ͨ͠ αϯϓϧαΠζେ αϯϓϧαΠζখ ޡࠩେ ޡࠩখ errP (h) ℰP
(h) = errP (h) − inf h*:Մଌؔ errP (h*) inf h*:Մଌؔ errP (h*) 𝔼 [ℰP (h)] ≤ U(n) n
Ұகੑ w༨ޡ͕ࠩαϯϓϧαΠζແݶେͷ࣌ʹʹऩଋ wਖ਼֬ʹͲΜͳʹରͯ͠ˢ͕Γཱͭ͜ͱ αϯϓϧαΠζେ αϯϓϧαΠζখ Ұகੑ͋Γ Ұகੑͳ͠ ޡࠩେ ޡࠩখ n
సҠֶशͷֶशཧ wߏஙͨ͠ΞϧΰϦζϜʹ͍ͭͯ༨ޡࠩͱιʔεͷαϯϓ ϧαΠζ ͱλʔήοτͷαϯϓϧαΠζ ͷؔΛ໌Β ͔ʹ͍ͨ͠ nP nQ
సҠֶशͷֶशཧ wߏஙͨ͠ΞϧΰϦζϜʹ͍ͭͯ༨ޡࠩͱιʔεͷαϯϓ ϧαΠζ ͱλʔήοτͷαϯϓϧαΠζ ͷؔΛ໌Β ͔ʹ͍ͨ͠ nP nQ ͲΕ͚ͩιʔεͷσʔλΛ׆༻Ͱ͖͔ͨʁ
సҠֶशͷֶशཧ wߏஙͨ͠ΞϧΰϦζϜʹ͍ͭͯ༨ޡࠩͱιʔεͷαϯϓ ϧαΠζ ͱλʔήοτͷαϯϓϧαΠζ ͷؔΛ໌Β ͔ʹ͍ͨ͠ nP nQ ιʔεαϯϓϧαΠζେ ιʔεαϯϓϧαΠζখ
λʔήοτޡࠩେ λʔήοτޡࠩখ nP errQ (h) ℰQ (h) = errQ (h) − inf h*:Մଌؔ errQ (h*) inf h*:Մଌؔ errQ (h*) 𝔼 [ℰQ (h)] ≤ U(nP , nQ )
ιʔεαϯϓϧαΠζʹର͢ΔҰகੑ w༨ޡ͕ࠩιʔεαϯϓϧαΠζແݶେͷ࣌ʹʹऩଋ wਖ਼֬ʹͲΜͳʹରͯ͠ˢ͕Γཱͭ͜ͱ Ұகੑ͋Γ Ұகੑͳ͠ ιʔεαϯϓϧαΠζେ ιʔεαϯϓϧαΠζখ λʔήοτޡࠩେ λʔήοτޡࠩখ nP
ιʔεαϯϓϧαΠζʹର͢ΔҰகੑ w༨ޡ͕ࠩιʔεαϯϓϧαΠζແݶେͷ࣌ʹʹऩଋ wਖ਼֬ʹͲΜͳʹରͯ͠ˢ͕Γཱͭ͜ͱ Ұகੑ͋Γ Ұகੑͳ͠ ιʔεαϯϓϧαΠζେ ιʔεαϯϓϧαΠζখ λʔήοτޡࠩେ λʔήοτޡࠩখ nP
ιʔεαϯϓϧΛֶͬͯश͕Ͱ͖͍ͯΔ ˠసҠͷޭ
γϑτ ֶशΞϧΰϦζϜ ྨث f( )=Ҝࢠ ιʔεσʔλ λʔήοτσʔλ ιʔεσʔλͱ༧ଌ࣌ͷσʔλ͕ શ͘ҟͳΔͱ༧ଌͰ͖ͳ͍ ιʔεͱλʔήοτԿ͔͠ΒͷҙຯͰࣅ͍ͯΔඞཁ͕͋Γ
0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ
$PWBSJBUF4IJGU ιʔε λʔήοτ ྨنଇಉҰ ೖྗσʔλҟͳΔ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ
$PWBSJBUF4IJGU ιʔε λʔήοτ ྨنଇಉҰ ೖྗσʔλҟͳΔ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ
ྨنଇ͕ಉ͡ ˠιʔε͚ͩͰྨ͕ޭ͢Δ ˠҰகੑʹసҠͷޭ
$PWBSJBUF4IJGU ιʔε λʔήοτ PX QX PY|X QY|X PX ≠
QX PY|X (Y = 1|X) = QY|X (Y = 1|X) = η(X) $PWBSJBUFTIJGUԾఆ η(X) = 1 2
طଘͷཧత݁Ռ
ؒڑΛͬͨ൚ԽޡࠩʹΑΔ্ք #FO%BWJEFUBM 1BSLFUBM "NJOJBOFUBM ʜ wཧղੳͷඪ 𝔼
[ℰQ (h)] ≤ U(nP , nQ ) λʔήοτͰଌͬͨࠩޡࠩ
ؒڑΛͬͨ൚ԽޡࠩʹΑΔ্ք #FO%BWJEFUBM 1BSLFUBM "NJOJBOFUBM ʜ w൚ԽޡࠩղੳΛ௨ͨࠩ͠ޡࠩͷ্ք 𝔼
[ℰQ (h)] ≤ errP,nP (h) + d(PX , QX ) + n−c P ιʔεͷܦݧޡࠩerrP,nP (h) = 1 nP nP ∑ i=1 1{h(Xi ) ≠ Yi } ؒڑ ιʔεͷܦݧޡࠩ ͕ࣅ͍ͯΔ΄ͲసҠֶश্͕ख͍͘͘ ݟ͕ͨࣅ͍ͯΔ
ؒڑΛͬͨ൚ԽޡࠩʹΑΔ্ք #FO%BWJEFUBM 1BSLFUBM "NJOJBOFUBM ʜ w൚ԽޡࠩղੳΛ௨ͨࠩ͠ޡࠩͷ্ք 𝔼
[ℰQ (h)] ≤ errP,nP (h) + d(PX , QX ) + n−c P ιʔεͷܦݧޡࠩerrP,nP (h) = 1 nP nP ∑ i=1 1{h(Xi ) ≠ Yi } ؒڑ ιʔεͷܦݧޡࠩ ͕ࣅ͍ͯΔ΄ͲసҠֶश্͕ख͍͘͘ ݟ͕ͨࣅ͍ͯΔ ຊʹʁ
ؒڑΛͬͨ൚ԽޡࠩʹΑΔ্ք #FO%BWJEFUBM 1BSLFUBM "NJOJBOFUBM ʜ ʹͰ͖ͳ͍ ͜ΕΒͷ্քͰαϯϓϧαΠζʹର͢ΔҰகੑΛࣔͤͳ͍
w൚ԽޡࠩղੳΛ௨ͨࠩ͠ޡࠩͷ্ք 𝔼 [ℰQ (h)] ≤ errP,nP (h) + d(PX , QX ) + n−c P
֬ൺΛ্ͬͨք ,QPUVGF .BFUBM 'FOHFUBM w֬ൺ wֶशΞϧΰϦζϜ ρ(x)
= dQX dPX (x) h = arg minh 1 nP ∑nP i=1 ρ(Xi )ℓ(h, (Xi , Yi )) ιʔε λʔήοτ PX QX ͍ॏΈ ߴ͍ॏΈ λʔήοτͬΆ͍σʔλΛ ߴ͘ධՁ͢Δ
֬ൺΛ্ͬͨք ,QPUVGF .BFUBM 'FOHFUBM w֬ൺ wֶशΞϧΰϦζϜ ρ(x)
= dQX dPX (x) h = arg minh 1 nP ∑nP i=1 ρ(Xi )ℓ(h, (Xi , Yi )) 𝔼 [ℰQ (h)] ≤ C ( ln(nP ) nP ) c ҰகੑΛ͍ࣔͤͯΔʁ
֬ൺΛ্ͬͨք ,QPUVGF .BFUBM 'FOHFUBM w֬ൺ wֶशΞϧΰϦζϜ ρ(x)
= dQX dPX (x) h = arg minh 1 nP ∑nP i=1 ρ(Xi )ℓ(h, (Xi , Yi )) 𝔼 [ℰQ (h)] ≤ C1 ( ln(nP ) nP ) c1 + C2 n−c2 Q ͷਪఆʹҰகੑΛ ્͢Δ߲͕ݱΕΔ ρ ֶशʹ֬ൺΛ͍ͬͯΔ ࣮ࡍʹಘΒΕͳ͍ ͜ΕΒͷ্քͰαϯϓϧαΠζʹର͢ΔҰகੑΛࣔͤͳ͍
ڑۭؒϕʔεؒྨࣅʹΑΔ্ք ,QPUVGFFUBM 1BUIBLFUBM (BMCSBJUIFUBM ڑ্ۭؒͷٿΛͱʹͨ͠ྨࣅ 1BUIBLFUBM
wڑۭؒ wܘ ͷٿ ( 𝒳 , ρ) r Bρ (x, r) = {x′  ∈ 𝒳 : ρ(x, x′  ) ≤ r} ΔPMW (P, Q; r) = ∫ 𝒳 1 PX (B(x, r)) QX (dx) ͷ࣌ ҰகੑΛ࣋ͭΞϧΰϦζϜΛߏங ΔPMW (P, Q; r) = O(r−τ) (τ < ∞) 𝔼 [ℰQ (h)] ≤ Cn−c P (c > 0) ࣮ࡍ 1BUIBLFUBM ճؼઃఆͰ͋Δ͕ɼ্هྨࣅྨʹద༻Մೳʢຊจʣ
ڑϕʔεؒྨࣅʹΑΔ্ք ,QPUVGFFUBM 1BUIBLFUBM (BMCSBJUIFUBM ڑ্ۭؒͷٿΛͱʹͨ͠ྨࣅ ΔPMW
(P, Q; r) = ∫ 𝒳 1 PX (B(x, r)) QX (dx) ׂΓࢉ͕ى͜ΔՄೳੑ ιʔε PX QX λʔήοτ ॏͳ͍ͬͯΔʢઈର࿈ଓʣ ˠׂى͜Βͳ͍
ڑϕʔεؒྨࣅʹΑΔ্ք ,QPUVGFFUBM 1BUIBLFUBM (BMCSBJUIFUBM ڑ্ۭؒͷٿΛͱʹͨ͠ྨࣅ ΔPMW
(P, Q; r) = ∫ 𝒳 1 PX (B(x, r)) QX (dx) ׂΓࢉ͕ى͜ΔՄೳੑ ιʔε PX QX λʔήοτ ͣΕ͍ͯΔʢඇઈର࿈ଓʣ ˠׂ͕ى͜Δʂ
ڑϕʔεؒྨࣅʹΑΔ্ք ,QPUVGFFUBM 1BUIBLFUBM (BMCSBJUIFUBM ڑ্ۭؒͷٿΛͱʹͨ͠ྨࣅ ΔPMW
(P, Q; r) = ∫ 𝒳 1 PX (B(x, r)) QX (dx) ׂΓࢉ͕ى͜ΔՄೳੑ ιʔε PX QX λʔήοτ ͣΕ͍ͯΔʢඇઈର࿈ଓʣ ˠׂ͕ى͜Δʂ ඇઈର࿈ଓͷঢ়ଶͰαϯϓϧαΠζʹର͢ΔҰகੑΛࣔͤͳ͍
ݱ࣮ੈքͰͷඇઈର࿈ଓੑ wྫ0 ff i DF)PNFEBUBTFU wͭͷυϝΠϯ Ξʔτ ΫϦοϓΞʔτ ϓϩμΫτ ϦΞϧ
wͷΧςΰϦ 0 ffi DF)PNF%BUBTFU)7FOLBUFTXBSBFUBM%FFQIBTIJOHOFUXPSLGPSVOTVQFSWJTFEEPNBJOBEBQUBUJPO$713 QQ ҟͳΔυϝΠϯͰग़ݱ͠ͳ͍ը૾͕͋Δˠඇઈର࿈ଓ
طଘݚڀͷ·ͱΊͱຊจͷߩݙ ߩݙ wඇઈର࿈ଓͰ͋ͬͨͱͯ͠ιʔεʹର͢ΔҰகੑΛࣔͤ ΔཧΛߏங wڑۭؒϕʔεͷཧΛ౷ҰతʹٞͰ͖Δํ๏Λߏங ͠ɼఏҊ͢ΔཧͷΑΓૣ͍ऩଋͷୡΛࣔ͢ ؒڑ ֬ൺ ڑۭؒϕʔε ຊݚڀ
ιʔεҰகੑ ✔ ✔ ඇઈର࿈ଓ ✔ ✔
ຊݚڀͷ݁Ռ
ͬͨ͜ͱ w৽͍͠ٿΛͱʹͨ͠ྨࣅΛఏҊ Δ 𝒱 (P, Q; r) = ∫ 𝒳
inf x′  ∈ 𝒱 (x) 1 PX (B(x′  , r)) QX (dx) ۙू߹ 𝒱 (x) ͷ࣌ ҰகੑΛ࣋ͭΞϧΰϦζϜΛߏங Δ 𝒱 (P, Q; r) = O(r−τ) (τ < ∞) *O fi NVNΛऔΔ͜ͱͰׂΓࢉΛ ͋ΔఔճආՄೳ
//ΞϧΰϦζϜ k wιʔεʴλʔήοταϯϓϧΛ׆༻ͨ͠ //ྨث k (X, Y)P (X, Y)Q ιʔεαϯϓϧ
λʔήοταϯϓϧ (X, Y) ݁߹ ςετೖྗX (X(1) , Y(1) ), . . . , (X(k) , Y(k) ) ͱڑ͕͍ۙ ݸΛநग़ X k ̂ ηk (X) = 1 k k ∑ i=1 Y(i) ̂ hk (X) = 1 { ̂ ηk (X) ≥ 1 2}
λʔήοτ ͷ͠͞ Q wλʔήοταϯϓϧͷΈͰͷྨͷ͠͞ͷԾఆ w4NPPUIOFTT /PJTFDPOEJUJPO w4NPPUIOFTT ͷ)ÖMEFS࿈ଓੑ
w/PJTFDPOEJUJPO 5TZCBLPWϊΠζ݅ η |η(x) − η(x′  )| ≤ Cα ρα(x, x′  ) QX (0 < |η(X)− 1 2 | ≤ t) ≤ Cβ tβ X ϥϕϧ͕ ϥϕϧ͕ η(X) 1 2 1 ϊΠζͷେ͖͞ ʢؒҧͬͨϥϕϧ͕ಘΒΕΔ֬ʣ େ͖͍ϊΠζك ۙ͘ͷϥϕϧಉ͡
ۙू߹ w ͷϥϕϧΛ༧ଌ͢Δͱ͖ϥϕϧ͕มΘΒͳ͍ۙ ͷϥϕϧΛ༧ଌͨ݁͠ՌΛͬͯྑ͍ X X′  𝒱 (x) =
{ x′  ∈ 𝒳 : 2Cα ρα(x, x′  ) < η(x) − 1 2 } X 𝒱 (X) ڥքΛ͑ͳ͍͙Β͍ͷ େ͖͞ͷٿ
సҠࢦɾࣗݾࢦ wڑۭؒϕʔεྨࣅ w Λͬͨ ͷಛ ͱ ͷಛ Δ(P, Q;
r) Δ (P, Q) τ Q ψ 𝔼 [ℰQ (h)] ≤ U(nP , nQ ) λʔήοτͰଌͬͨࠩޡࠩ wཧղੳͷඪ 𝔼 [ℰQ (h)] ≤ C (nc(τ) P + nc(ψ) Q ) −1 ͷ߲ͱ ͷ߲ͷ͠ࢉ nP nQ Λେ͖͘͢Εʹऩଋ ˠҰகੑ nP
సҠࢦɾࣗݾࢦ wڑۭؒϕʔεྨࣅ w Λͬͨ ͷಛ ͱ ͷಛ సҠࢦ
ࣗݾࢦ Δ(P, Q; r) Δ (P, Q) τ Q ψ Δ τ sup r∈(0,D 𝒳 ( r D 𝒳 ) τ Δ(P, Q; r) ≤ C Δ ψ sup r∈(0,D 𝒳 ( r D 𝒳 ) ψ Δ(Q, Q; r) ≤ C Δ(P, Q; r) = O(r−τ) Δ(Q, Q; r) = O(r−ψ)
ओ݁Ռ ʢఆཧʣ ࿈ଓੑɼ ϊΠζ͕݅Γཱͪɼ ࣗݾࢦ Λ࣋ͭɽ సҠࢦ
Λ࣋ͭɽ //ྨثҎ Լͷ্քΛ࣋ͭɽ Q α β Δ 𝒱 ψ (P, Q) Δ 𝒱 τ k C (n 1 + β 2 + β +max{1,τ/α} P + n 1 + β 2 + β +max{1,ψ/α} Q ) −1
ओ݁Ռ w௨ৗઃఆͷ࠷దϨʔτ ʢ ࣍ݩʣ "VEJCFSU FUBM w࣮ࡍ ࣍ݩͱࣅͨΑ͏ͳੑ࣭Λ࣋ͭ
n− 1 + β 2 + β + d/α d ψ ʢఆཧʣ ࿈ଓੑɼ ϊΠζ͕݅Γཱͪɼ ࣗݾࢦ Λ࣋ͭɽ సҠࢦ Λ࣋ͭɽ //ྨثҎ Լͷ্քΛ࣋ͭɽ Q α β Δ 𝒱 ψ (P, Q) Δ 𝒱 τ k C (n 1 + β 2 + β +max{1,τ/α} P + n 1 + β 2 + β +max{1,ψ/α} Q ) −1 సҠࢦ ࣗݾࢦ
సҠࢦɾࣗݾࢦʹΑΔطଘ݁Ռͷ࠶ղऍ wطଘͷ݁ՌҟͳΔ Λ͍ͬͯΔͱղऍͰ͖Δ 1BUIBLFUBM ,QPUVGFFUBM
Δ ΔPMW (P, Q; r) = ∫ 𝒳 1 PX (B(x, r)) QX (dx) ΔDM (Q, Q; r) = sup x∈ 𝒳 Q 1 QX (B(x, r)) ΔBCN (Q, Q; r) = 𝒩 ( 𝒳 Q , ρ, r) ΔKM (Q, Q; r) = sup x∈ 𝒳 Q QX (B(x, r)) PX (B(x, r)) ඃෳ
సҠࢦɾࣗݾࢦʹΑΔطଘ݁Ռͷ࠶ղऍ ʢఆཧʣ ࿈ଓੑɼ ϊΠζ͕݅Γཱͭɽ ʹ͍ͭ ͯҎԼͷ͍ͣΕ͔͕Γཱͭɽ ͕ ࣗݾࢦ
ɼ ͕ సҠࢦ Λ࣋ͭ ͕ PS ࣗݾࢦ ɼ ͕ సҠࢦ Λ͔࣋ͭͭ ͜ͷ࣌ //ྨثओఆཧͱಉ্͡քΛ࣋ͭɽͭ·Γɼ Q α β (P, Q) Q ΔPMW ψ (P, Q) ΔPMW τ Q ΔDM ΔBCN ψ (P, Q) ΔKM τ − ψ τ ≥ ψ k C (n 1 + β 2 + β +max{1,τ/α} P + n 1 + β 2 + β +max{1,ψ/α} Q ) −1 Λൺֱ͢Ε্քͷྑ͠ѱ͕͠ൺֱͰ͖Δ Δ
ͷൺֱ Δ ʢఆཧʣҙͷ ʹ͍ͭͯ ͕࣋ͭ࠷খͷ సҠࢦɾࣗݾࢦ w
ఏҊ͍ͯ͠Δ ͷసҠࢦɾࣗݾࢦ͕Ұ൪খ͍͞ w ˠҰ൪ૣ͍ऩଋΛ্ࣔ͢ք͕ಘΒΕΔ (P, Q) τΔ 𝒱 ≤ τΔPMW ≤ τΔKM + min{ψΔDM , ψΔDM } ψΔ 𝒱 ≤ τΔPMW ≤ min{ψΔDM , ψΔDM } τΔ , ψΔ (P, Q) Δ Δ 𝒱
࣮ݧ ͷਓσʔλͷ࣮ݧΛ࣮ࢪ wӈਤͷɾճؼؔ w ධՁࢦඪ wαΠζͷςετσʔληοτ Ͱܭࢉͨ͠༨ޡࠩ 𝒳 =
ℝ nP ∈ {28,29, . . . ,218}, nQ = 10 ੨ιʔεͷີؔ ᒵλʔήοτͷີؔ αϙʔτ͕ҟͳΔྖҬ ճؼؔ BMQIB CBUB UBV QTJ 1.8 PS BMQIB ♾ 0VS PS BMQIB PS ඇઈର࿈ଓΑΓ
݁Ռ w1.8PVSཧόϯυͱ ͖͕ಉ͡ wόϯυλΠτ w1.8ޡ͕ࠩݮΒͳ͍ wҰகੑ͕ͳ͍ w0VSޡ͕ࠩݮ͍ͬͯΔ wҰகੑΛࣔ͢ α =
0.5,τ = 2.0 α = 0.25,τ = 2.0 ιʔεαϯϓϧαΠζ ιʔεαϯϓϧαΠζ
·ͱΊ w$PWBSJBUFTIJGUԼͰιʔεαϯϓϧαΠζʹର͢ΔҰகੑ ΛࣔͤΔཧΛߏங w͜ͷঢ়گԼͰͷసҠͷޭΛࣔ͢ wಛʹۙใΛ׆༻͠ඇઈର࿈ଓͳঢ়گͰҰகੑΛࣔ͢ ͜ͱ͕Մೳ .JUTVIJSP'VKJLBXB :PIFJ"LJNPUP +VO4BLVNB BOE
,B[VUP'VLVDIJ)BSOFTTJOHUIF1PXFSPG7JDJOJUZ *OGPSNFE"OBMZTJTGPS$MBTTJ fi DBUJPOVOEFS$PWBSJBUF 4IJGUIUUQTBSYJWPSHBCT